Examples

Today, we’re going to explore linear inequalities. These are different from equations because they represent a range of values instead of a single solution. Can anyone tell me an example of a situation where you might use an inequality?

Like when there’s a speed limit? It’s not just one speed, but a maximum.

Exactly! Speed limits are a perfect example. So when we write an inequality like v ≤ 60, it means the velocity is less than or equal to 60 mph. What inequality signs do we use?

Less than, greater than, and less than or equal to?

Yes, great! Remember: <, >, ≤, and ≥. A useful mnemonic is 'The less than sign opens towards the left.'

Now, let's solve a linear inequality: 2x - 3 < 5. Who can help me solve this?

First, we add 3 to both sides to get 2x < 8.

Correct! What’s the next step?

We divide by 2 to find x < 4.

Excellent! So how do we graph this on a number line?

We place an open circle at 4 and shade to the left.

Right! Remember, the open circle shows that 4 is not included in the solution. Let's summarize: whenever you divide or multiply by a negative, you reverse the inequality sign!

Next, let’s graph a linear inequality in two variables like x + y < 4. What’s our first step?

We rewrite it as y < -x + 4.

Correct! Now, how do we draw the boundary line?

We use a dashed line because it's a < inequality.

Yes, perfect! Now we test the point (0, 0). What do we get?

0 + 0 < 4 is true, so we shade the area that includes (0, 0).

Excellent work! Shading correctly is crucial because it helps visualize the solution set.

Now, let’s talk about systems of linear inequalities. If we have x + y ≤ 6 and x ≥ 2, what would our solution look like?

We need to graph both inequalities and see where the shaded areas overlap.

Correct! Let’s graph both inequalities on the same plane. What do we start with?

We graph x + y ≤ 6 with a solid line and shade below it.

And then for x ≥ 2?

We draw a solid vertical line at x = 2 and shade to the right.

Yes! The solution is where these shaded areas overlap. Don't forget to check if points within this region satisfy both inequalities.

Finally, let’s apply what we've learned to a word problem. A student has $20 to spend on snacks, and each snack costs $2. How do we represent this situation mathematically?

We can use the inequality 2x ≤ 20, where x is the number of snacks.

Exactly! Now, how do we solve for x?

We divide by 2 to get x ≤ 10.

Right! This means the student can buy at most 10 snacks. Remember, word problems can often be translated into inequalities nicely!